WEAK FORMULATION OF MIXED STATE EQUATION ANDBOUNDARY VALUE PROBLEM OF LAMINATEDCYLINDRICAL SHELL

Weak formulation of mixed state equations including boundary conditions are presented in a cylindrical coordinate system by introducing Hellinger-Reissner variational principle. Analytical solutions are obtained for laminated cylindrical shell by means of state space method. The present study extends and unifies the solution of laminated shells.

Static Analysis of Doubly-Curved Shell Structures of Smart Materials and Arbitrary Shape Subjected to General Loads Employing Higher Order Theories and Generalized Differential Quadrature Method

The article proposes an Equivalent Single Layer(ESL)formulation for the linear static analysis of arbitrarily-shaped shell structures subjected to general surface loads and boundary conditions.A parametrization of the physical domain is provided by employing a set of curvilinear principal coordinates.The generalized blendingmethodology accounts for a distortion of the structure so that disparate geometries can be considered.Each layer of the stacking sequence has an arbitrary orientation and is modelled as a generally anisotropic continuum.In addition,re-entrant auxetic three-dimensional honeycomb cells with soft-core behaviour are considered in the model.The unknown variables are described employing a generalized displacement field and pre-determined through-the-thickness functions assessed in a unified formulation.Then,a weak assessment of the structural problem accounts for shape functions defined with an isogeometric approach starting fromthe computational grid.Ageneralizedmethodology has been proposed to define two-dimensional distributions of static surface loads.In the same way,boundary conditions with three-dimensional features are implemented along the shell edges employing linear springs.The fundamental relations are obtained from the stationary configuration of the total potential energy,and they are numerically tackled by employing the Generalized Differential Quadrature(GDQ)method,accounting for nonuniform computational grids.In the post-processing stage,an equilibrium-based recovery procedure allows the determination of the three-dimensional dispersion of the kinematic and static quantities.Some case studies have been presented,and a successful benchmark of different structural responses has been performed with respect to various refined theories.

Application of Modified Benders Decomposition to Single-Stage Multi-Commodity Multi-Period Warehouse Location Problem: An Empirical Investigation

In the past, researchers have applied Bender’s decomposition to distribution problem and used feasibility constraint to speed up the performance of Bender’s decomposition. Further, the application of Branch and Bound to single-stage multi-commodity single-period warehouse location problem (SSMCSPWLP) with strong constraints has shown that they are more effective. It was also shown in the previous research (in the context of Branch and Bound Methodology) that hybrid formulation for the single-stage single-period multi-commodity warehouse location problem yielded superior results. In this paper we apply Benders’ decomposition to strong and weak formulations of single-stage multi-commodity multi-period warehouse location problem (SSMCMPWLP). As suggested in the previous literature we put feasibility constraints in the pure integer sub- problem to speed up the performance of Benders’ decomposition. We also develop an additional cut (constraint that is again added to pure integer sub-problem) and show that it further speeded up Benders’ Decomposition. This research led to the possibility of applying Benders’ Decomposition to the hybrid formulation of SSMCMPWLP in future.

ERROR ESTIMATION FOR NUMERICAL METHODS USING THE ULTRA WEAK VARIATIONAL FORMULATION IN MODEL OF NEAR FIELD SCATTERING PROBLEM

In this paper, we investigate the use of ultra weak variational formulation to solve a wave scattering problem in near field optics. In order to capture the sub-scale features of waves, we utilize evanescent wave functions together with plane wave functions to approximate the local properties of the field. We analyze the global convergence and give an error estimation of the method. Numerical examples are also presented to demonstrate the effectiveness of the strategy.

TheUltraWeakVariational FormulationUsing Bessel Basis Functions

We investigate the ultra weak variational formulation(UWVF)of the 2-D Helmholtz equation using a new choice of basis functions.Traditionally the UWVF basis functions are chosen to be plane waves.Here,we instead use first kind Bessel functions.We compare the performance of the two bases.Moreover,we show that it is possible to use coupled plane wave and Bessel bases in the same mesh.As test cases we shall consider propagating plane and evanescent waves in a rectangular domain and a singular 2-D Helmholtz problem in an L-shaped domain.

Excitonic Eigenstates of Disordered Semiconductor Quantum Wires:Adaptive Wavelet Computation of Eigenvalues for the Electron-Hole Schrodinger Equation

A novel adaptive approach to compute the eigenenergies and eigenfunctions of the two-particle(electron-hole)Schrodinger equation including Coulomb attraction is presented.As an example,we analyze the energetically lowest exciton state of a thin one-dimensional semiconductor quantum wire in the presence of disorder which arises from the non-smooth interface between the wire and surrounding material.The eigenvalues of the corresponding Schrodinger equation,i.e.,the onedimensional exciton Wannier equation with disorder,correspond to the energies of excitons in the quantum wire.The wavefunctions,in turn,provide information on the optical properties of the wire.We reformulate the problem of two interacting particles that both can move in one dimension as a stationary eigenvalue problem with two spacial dimensions in an appropriate weak form whose bilinear form is arranged to be symmetric,continuous,and coercive.The disorder of the wire is modelled by adding a potential in the Hamiltonian which is generated by normally distributed random numbers.The numerical solution of this problem is based on adaptive wavelets.Our scheme allows for a convergence proof of the resulting scheme together with complexity estimates.Numerical examples demonstrate the behavior of the smallest eigenvalue,the ground state energies of the exciton,together with the eigenstates depending on the strength and spatial correlation of disorder.

A Solver for Helmholtz System Generated by the Discretization of Wave Shape Functions

An interesting discretization method for Helmholtz equations was introduced in B.Despres[1].This method is based on the ultra weak variational formulation(UWVF)and the wave shape functions,which are exact solutions of the governing Helmholtz equation.In this paper we are concerned with fast solver for the system generated by the method in[1].We propose a new preconditioner for such system,which can be viewed as a combination between a coarse solver and the block diagonal preconditioner introduced in[13].In our numerical experiments,this preconditioner is applied to solve both two-dimensional and three-dimensional Helmholtz equations,and the numerical results illustrate that the new preconditioner is much more efficient than the original block diagonal preconditioner.